125 research outputs found
The Block Spin Renormalization Group Approach and Two-Dimensional Quantum Gravity
A block spin renormalization group approach is proposed for the dynamical
triangulation formulation of two-dimensional quantum gravity. The idea is to
update link flips on the block lattice in response to link flips on the
original lattice. Just as the connectivity of the original lattice is meant to
be a lattice representation of the metric, the block links are determined in
such a way that the connectivity of the block lattice represents a block
metric. As an illustration, this approach is applied to the Ising model coupled
to two-dimensional quantum gravity. The correct critical coupling is
reproduced, but the critical exponent is obscured by unusually large finite
size effects.Comment: 10 page
Ising Model Coupled to Three-Dimensional Quantum Gravity
We have performed Monte Carlo simulations of the Ising model coupled to
three-dimensional quantum gravity based on a summation over dynamical
triangulations. These were done both in the microcanonical ensemble, with the
number of points in the triangulation and the number of Ising spins fixed, and
in the grand canoncal ensemble. We have investigated the two possible cases of
the spins living on the vertices of the triangulation (``diect'' case) and the
spins living in the middle of the tetrahedra (``dual'' case). We observed phase
transitions which are probably second order, and found that the dual
implementation more effectively couples the spins to the quantum gravity.Comment: 11 page
Bubble divergences: sorting out topology from cell structure
We conclude our analysis of bubble divergences in the flat spinfoam model. In
[arXiv:1008.1476] we showed that the divergence degree of an arbitrary
two-complex Gamma can be evaluated exactly by means of twisted cohomology.
Here, we specialize this result to the case where Gamma is the two-skeleton of
the cell decomposition of a pseudomanifold, and sharpen it with a careful
analysis of the cellular and topological structures involved. Moreover, we
explain in detail how this approach reproduces all the previous powercounting
results for the Boulatov-Ooguri (colored) tensor models, and sheds light on
algebraic-topological aspects of Gurau's 1/N expansion.Comment: 19 page
Quantum Gravity Vacuum and Invariants of Embedded Spin Networks
We show that the path integral for the three-dimensional SU(2) BF theory with
a Wilson loop or a spin network function inserted can be understood as the
Rovelli-Smolin loop transform of a wavefunction in the Ashtekar connection
representation, where the wavefunction satisfies the constraints of quantum
general relativity with zero cosmological constant. This wavefunction is given
as a product of the delta functions of the SU(2) field strength and therefore
it can be naturally associated to a flat connection spacetime. The loop
transform can be defined rigorously via the quantum SU(2) group, as a spin foam
state sum model, so that one obtains invariants of spin networks embedded in a
three-manifold. These invariants define a flat connection vacuum state in the
q-deformed spin network basis. We then propose a modification of this
construction in order to obtain a vacuum state corresponding to the flat metric
spacetime.Comment: 15 pages, revised version to appear in Class. Quant. Gra
Smooth Random Surfaces from Tight Immersions?
We investigate actions for dynamically triangulated random surfaces that
consist of a gaussian or area term plus the {\it modulus} of the gaussian
curvature and compare their behavior with both gaussian plus extrinsic
curvature and ``Steiner'' actions.Comment: 7 page
2D Conformal Field Theories and Holography
It is known that the chiral part of any 2d conformal field theory defines a
3d topological quantum field theory: quantum states of this TQFT are the CFT
conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT
relation exists also for the full CFT. The 3d topological theory that arises is
a certain ``square'' of the chiral TQFT. Such topological theories were studied
by Turaev and Viro; they are related to 3d gravity. We establish an
operator/state correspondence in which operators in the chiral TQFT correspond
to states in the Turaev-Viro theory. We use this correspondence to interpret
CFT correlation functions as particular quantum states of the Turaev-Viro
theory. We compute the components of these states in the basis in the
Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we
obtain is a generalization of the Verlinde formula. The later is obtained from
our expression for a zero colored graph. Our results give an interesting
``holographic'' perspective on conformal field theories in 2 dimensions.Comment: 29+1 pages, many figure
An Effective Model for Crumpling in Two Dimensions?
We investigate the crumpling transition for a dynamically triangulated random
surface embedded in two dimensions using an effective model in which the
disordering effect of the variables on the correlations of the normals is
replaced by a long-range ``antiferromagnetic'' term. We compare the results
from a Monte Carlo simulation with those obtained for the standard action which
retains the 's and discuss the nature of the phase transition.Comment: 5 page
A Potts/Ising Correspondence on Thin Graphs
We note that it is possible to construct a bond vertex model that displays
q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary
topology, which we denote as ``thin'' random graphs in contrast to the fat
graphs of the planar diagram expansion.
Since the four vertex model in question also serves to describe the critical
behaviour of the Ising model in field, the formulation reveals an isomorphism
between the Potts and Ising models on thin random graphs. On planar graphs a
similar correspondence is present only for q=1, the value associated with
percolation.Comment: 6 pages, 5 figure
Ising Model on Networks with an Arbitrary Distribution of Connections
We find the exact critical temperature of the nearest-neighbor
ferromagnetic Ising model on an `equilibrium' random graph with an arbitrary
degree distribution . We observe an anomalous behavior of the
magnetization, magnetic susceptibility and specific heat, when is
fat-tailed, or, loosely speaking, when the fourth moment of the distribution
diverges in infinite networks. When the second moment becomes divergent,
approaches infinity, the phase transition is of infinite order, and size effect
is anomalously strong.Comment: 5 page
Quantum Adiabatic Algorithm and Large Spin Tunnelling
We provide a theoretical study of the quantum adiabatic evolution algorithm
with different evolution paths proposed in [E. Farhi, et al.,
arXiv:quant-ph/0208135]. The algorithm is applied to a random binary
optimization problem (a version of the 3-Satisfiability problem) where the
n-bit cost function is symmetric with respect to the permutation of individual
bits. The evolution paths are produced, using the generic control Hamiltonians
H(\tau) that preserve the bit symmetry of the underlying optimization problem.
In the case where the ground state of H(0) coincides with the totally-symmetric
state of an n-qubit system the algorithm dynamics is completely described in
terms of the motion of a spin-n/2.
We show that different control Hamiltonians can be parameterized by a set of
independent parameters that are expansion coefficients of H(\tau) in a certain
universal set of operators. Only one of these operators can be responsible for
avoiding the tunnelling in the spin-n/2 system during the quantum adiabatic
algorithm. We show that it is possible to select a coefficient for this
operator that guarantees a polynomial complexity of the algorithm for all
problem instances. We show that a successful evolution path of the algorithm
always corresponds to the trajectory of a classical spin-n/2 and provide a
complete characterization of such paths.Comment: 27 pages, 3 figure
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